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WO2025017158A1 - Method for estimating a flow profile in a shadowed region from ultrasonic signal data - Google Patents

Method for estimating a flow profile in a shadowed region from ultrasonic signal data Download PDF

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Publication number
WO2025017158A1
WO2025017158A1 PCT/EP2024/070471 EP2024070471W WO2025017158A1 WO 2025017158 A1 WO2025017158 A1 WO 2025017158A1 EP 2024070471 W EP2024070471 W EP 2024070471W WO 2025017158 A1 WO2025017158 A1 WO 2025017158A1
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physics
flow
data
loss function
machine learning
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French (fr)
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Steven Freear
Luzhen NIE
Elliott Smith
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University of Leeds
University of Leeds Innovations Ltd
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University of Leeds
University of Leeds Innovations Ltd
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    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B8/00—Diagnosis using ultrasonic, sonic or infrasonic waves
    • A61B8/06—Measuring blood flow
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B8/00—Diagnosis using ultrasonic, sonic or infrasonic waves
    • A61B8/08—Clinical applications
    • A61B8/0891—Clinical applications for diagnosis of blood vessels
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B8/00—Diagnosis using ultrasonic, sonic or infrasonic waves
    • A61B8/46—Ultrasonic, sonic or infrasonic diagnostic devices with special arrangements for interfacing with the operator or the patient
    • A61B8/461—Displaying means of special interest
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/045—Combinations of networks
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B8/00—Diagnosis using ultrasonic, sonic or infrasonic waves
    • A61B8/08—Clinical applications
    • A61B8/0833—Clinical applications involving detecting or locating foreign bodies or organic structures
    • A61B8/085—Clinical applications involving detecting or locating foreign bodies or organic structures for locating body or organic structures, e.g. tumours, calculi, blood vessels, nodules
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B8/00—Diagnosis using ultrasonic, sonic or infrasonic waves
    • A61B8/08—Clinical applications
    • A61B8/0883—Clinical applications for diagnosis of the heart

Definitions

  • the invention relates to a method, particularly a computer-implemented method for estimating a flow profile of a fluid flow in a vessel.
  • Ultrasonic flow imaging has been established as a front-line tool in diagnosing a range of diseases such as vessel stenosis and congenital heart diseases, among others. Its high frame rates (> tens of Hz) provide the unique advantage of accurately capturing complex flow patterns which could be clinically informative but cannot be readily achieved by other imaging modalities such as Magnetic Resonance Imaging (MRI) and Computed Tomography (CT).
  • MRI Magnetic Resonance Imaging
  • CT Computed Tomography
  • ultrasonic flow imaging has its specific difficulties in challenging situations, such as acoustic shadowing that obscures the visualization of the flow field and the related vessel morphology.
  • Stroke is one of the leading causes of mortality and disability globally together with cancer and ischaemic heart disease.
  • Atherosclerosis with a plaque as the main manifestation, is responsible for many stroke cases.
  • a biochemical signal triggered by the endothelium induces the repair with inflammatory cells and gathering of cholesterol, fats and other substances at the damaged site, with the formation of a fatty plaque protruding into the vessel lumen.
  • the vessel luminal diameter is subsequently reduced and eventually a stenosis occurs.
  • Ultrasound imaging has been the mainstay for diagnosing vessel stenosis noninvasively, and it has the advantages of being non-radioactive, bed-side accessible and affordable, compared to other imaging modalities.
  • the analysis of morphology (ratio-percent method) and Peak-Systolic-Velocity (PSV) values are two common approaches adopted to quantify the degree of stenosis by using ultrasound. And corresponding thresholds are used to evaluate the severity and inform the consequent diagnostic and therapeutic approach.
  • calcifications are present in many plaques, and the calcified plaques result in an acoustic shadow that obscures the vessel lumen, inhibiting the sonographer’s ability to obtain ultrasound measurements.
  • acoustic shadowing occurs in 14.7% of the participants.
  • Acoustic shadowing occurs when strongly-reflecting plaques on the vessel wall are insonified by ultrasound beams. This artefact is projected on regions beneath with low intensity echoes returned, causing significant problems in morphological and haemodynamic evaluation of the stenosis.
  • a physics-informed machine learning method allows an improved prediction of the flow in the shadowed region of the vessel, based on physical laws and properties that are taken into account by the physics-informed machine learning method.
  • Results of the method according to the first aspect reflect a more likely fluid flow in the vessel than other machine learning methods trained to predict the fluid flow in the shadowed region, as these methods may not rely on physical law and properties of the fluid and the geometry of the vessel. Particularly it is possible to also predict the temporal evolution of the fluid flow.
  • the flow profile is determined for at least two flow regions, wherein a first of the at least two flow regions is located upstream the shadowed region and a second of the at least two flow regions is located downstream of the shadowed region, particularly wherein the fluid flow flows from an upstream region to a downstream region of the shadowed region.
  • the up- and downstream regions are therefore defined by means of a net flow direction.
  • the up and/or the downstream region may comprise one or more vessels.
  • the flow regions are located adjacent to the shadowed region.
  • the shadowed region is detected automatically by the method.
  • the shadowed region extends along a net flow direction up to 5 mm, 10 mm, 20 mm, 50 mm, 100 mm, 150 mm, or even up to 300 mm.
  • the shadowed region or a vessel diameter in the shadowed region may be in the range of 0.5 mm to 50 mm, particularly in the range of 1 mm to 30 mm.
  • a fully-connected deep neural network is utilized to implement the physics-informed machine learning method.
  • the fully-connected deep neural network may take spatial and time coordinates, i.e. locations and time points, as inputs and predicts multidimensional velocity components and pressure.
  • the inputs are spatial coordinates x, y and the corresponding time coordinate t
  • the outputs are velocity components u, v, and pressure p.
  • there is one additional input z the third spatial coordinate, and one additional output w , the third velocity component.
  • the fully-connected deep neural network may comprise multiple hidden layers, with multiple neurons in each hidden layer.
  • a sinusoidal activation function may be used for each neuron. Note that other activation functions can be used, particularly as long as they have meaningful second-order derivatives for the determination of the physics loss, as described below.
  • the flow data are used to train the physics-informed machine learning method to reconstruct velocities at a plurality of locations and time points to match the velocities comprised in the flow profile based on a flow data loss function, particularly wherein the flow data loss function is indicative of deviations of the reconstructed velocities and the velocities comprised in the flow profile.
  • the location data may be sampled in and outside the shadowed region, wherein the location data are used to train the physics-informed machine learning method to reconstruct velocities and pressure for a plurality of locations and time points in the shadowed region applying a physics loss function.
  • the machine learning method is constrained to produce solutions that respect the underlying physics of the flow, even in the shadowed region.
  • training of the physics-informed machine learning method is facilitated by minimizing a composite error function comprising the flow data loss function and the physics loss function.
  • the flow data sampled outside the shadowed region are leveraged as boundary or initial conditions to guide the solution.
  • Traditional machine learning methods might produce nonphysical flow profiles in the shadowed region due to the lack of training data.
  • This physics-informed machine learning method may be trained to satisfy the governing equations universally across the domain. Thus, it can generalize to shadowed regions with no direct measurements, providing a structured way to reconstruct a coherent and physically plausible dense flow profile across the entire domain of interest, including the shadowed region.
  • the composite loss function and its corresponding composite loss value is minimized by a gradient descent method.
  • flow data are generated, particularly sampled from the flow profile determined for the flow regions, wherein the flow data comprise information on a velocity of the fluid for a plurality of locations at a plurality of associated time points.
  • the flow data therefore represent a spatially resolved fluid dynamics or flow profile in the vessel portions that are not shadowed, e.g. in the flow regions.
  • Various methods are available to determine a flow from the signal data. For example, it is possible to determine the flow profile by speckle tracking and vector Doppler methods.
  • the flow data are restricted to locations comprised by the flow regions.
  • the flow data may comprise information on a velocity for one or more locations in the flow regions at one or more time points.
  • training of the physics-informed machine learning method comprises providing information on locations and time points from the flow data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity, wherein a flow data loss function determines a flow data loss value indicative of a deviation of the velocities comprised in the flow data from velocities estimated by the physics-informed machine learning method.
  • flow data loss and the term “data loss” may be used synonymously.
  • the flow data comprising information on the flow profile is particularly well-suited for a physics-informed machine learning method, as fluid flow may be physically modelled by means of a dynamic information on the flow profile of the fluid.
  • This embodiment allows for example to train the physics-informed machine learning method to reconstruct velocities in regions outside the shadowed regions. This allows for matching the velocities and in turn contributes for example as constraints for a physics loss function that may be based on a Navier-Stokes equation.
  • training of the physics-informed machine-learning method comprises providing at least some locations and associated time points from the location data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity and an estimated pressure of the fluid at the location and the time point, wherein the estimated velocities and the estimated pressures are provided to a physics loss function comprising the Navier-Stokes equations, wherein the physics loss function determines a physics loss value indicative of a deviation of the velocities and the pressures estimated by the physics-informed machine-learning method from velocities and pressures that would solve the Navier-Stokes equations. Assigning the Navier-Stokes Equation to a physics loss function provides a suitable physics-informed and physics enforced loss function, such that estimations of the machine learning method are bound to physically meaningful flow profiles in the shadowed region.
  • training comprises a first stage, a second stage, and a third stage, wherein in the first stage, the data loss value is determined, in the second stage the physics loss value is determined, and in the third stage a composite loss value is determined from a composite loss function comprising the data loss function and the physics loss function, particularly wherein the composite loss function comprises a sum, particularly a weighted sum, of the data loss function and the physics loss function, wherein the composite loss function is minimized during training episodes.
  • This three-stage process allows to model a robust training process, wherein the flow data loss function and the physics loss function are separable and their contribution to the output estimate of the flow profile may be adjusted by the composite loss function.
  • the Navier-Stokes equations correspond to Navier-Stokes equations adjusted for incompressible fluids, conservation of momentum, and conservation of mass.
  • the flow profile is determined either three-dimensionally or two-dimensionally.
  • the physics loss function comprises physics residual functions, , fa, fa, fa, based on the terms of Navier-Stokes equations: wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, w are velocity components of the velocity estimated by the physics-informed machine learning method along three dimensions x, y, z, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, L physics , and the physics loss function is determined according to: wherein N physics denotes the number of locations and associated time points for which the physics loss function has been evaluated.
  • the physics loss function comprises physics residual functions, , fa, fa, based on the terms of Navier-Stokes equations: wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, are velocity components estimated by the physics-informed machine learning method along two dimensions x, y, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, L p h yS ics, ar
  • the data loss function and the data loss value, L data is determined according to: wherein u v, w are velocity components for location Xj, yj, Zj at time point ty of the velocity estimated by the physics-informed machine learning method and wherein “reference , “reference . Reference are the corresponding velocity components of the velocity of the provided flow data, wherein W data , denotes the number of locations and particularly associated time points for which the data loss function has been evaluated.
  • the data loss function and the data loss value, L data is determined according to: wherein u, v are velocity components for location x,-,yy at time point ty of the velocity estimated by the physics-informed machine learning method and wherein u reference , “reference are the corresponding velocity components of the velocity of the provided flow data, wherein N data , denotes the number of locations and particularly associated time points for which the data loss function has been evaluated.
  • the vessel is a blood vessel in a living body.
  • the shadowed region may be caused by a calcified vessel wall section.
  • the method is executed during an ultrasonic imaging examination of a patient, wherein the training of the physics- informed machine learning method is executed for each patient at least once.
  • sufficient training data may be acquired rapidly as the regions adjacent to the shadowed region provide a temporal flow profile that may repeat itself after each heartbeat, such that acquiring the training data is facilitated with one or a few heartbeats.
  • the duration of acquisition may be in the range of 0.5s to 10s.
  • a vessel boundary is determined at least for the flow regions, wherein a velocity of the determined flow profile is set to zero at the boundary and outside the flow regions and outside the shadowed region.
  • a flow profile in the shadowed region is determined from the estimated velocities using the trained physics-informed machine learning method by submitting corresponding spatial and time coordinates, e.g. location data and associated time points.
  • a time-resolved velocity and/or a pressure distribution for at least a portion of the shadowed region, particularly for the complete shadowed region is estimated by the trained physics-informed machine learning method.
  • the velocity and/or the pressure estimated by the trained physics-informed machine learning method is displayed, particularly wherein the estimated velocity and/or the estimated pressure is displayed in combination with the flow region and the shadowed region.
  • a sequence of velocities and/or pressures as estimated by the trained physics-informed machine learning method is generated, particularly in form of a movie, such that a temporal evolution of the velocity and/or the pressure, particularly a temporal evolution of the velocity distribution and/or the estimated pressure distribution is generated, wherein said sequence(s) is/are displayed together with the flow profile, particularly with the velocities in the flow regions, such that the flow regions and the shadowed region show a continuous flow profile evolving over time, particularly in form of a movie.
  • the flow profile in the flow regions is determined by means of tracking tracers comprised in the fluid.
  • Such tracers may comprise micro-bubbles.
  • the method comprises the step of segmenting the vessel, such that at least a boundary of the vessel, and more particularly that a lumen of the vessel is determined, wherein said step is executed together, particularly simultaneously with the step of estimating the flow profile.
  • a computer program comprises computer program code, that when executed on a computer causes the computer to execute the method according to one of the preceding claims.
  • the computer program may be stored on a non-transitory storage medium.
  • an ultrasonic measurement system comprises at least the following components:
  • An ultrasonic transducer head system configured to generate and record the ultrasonic signal data;
  • a computer connected to the ultrasonic transducer head system and configured to receive the ultrasonic signal data, characterized in that the computer is configured to execute the method according to the first aspect of the invention or the computer program according to the second aspect of the invention.
  • the method aims to reconstruct the missing flow profile and vessel boundary information under the acoustic shadow, with immediate applications to the analysis of vessel stenosis.
  • the inventors have found a surprising way to estimate a flow profile in a shadowed region, in which no flow data may be acquired from.
  • At the core of the invention is the physics-informed machine learning method, that by design generates solutions that obey or at least almost obey a physical principle.
  • the underlying physical principle comprises the principles or laws of fluid dynamics.
  • the machine learning method is biased towards estimations that are physical meaningful, i.e. the estimates conform to a high degree to the physical principle underlying the observed natural phenomenon, here for example a fluid flow through a vessel.
  • This inventive concept in particular extends to the use of one or more equations that are written such that they may serve as an error function for penalizing deviations from said equation, i.e. the physically expected solution.
  • the physics-informed machine-learning method comprises an error function that at least in part comprises the laws of fluid dynamics. This part is referred to as physics loss function in the context of the current specification.
  • the inventors realized that in the flow regions outside, and particularly adjacent to the shadowed region, information is present that can be used for preventing generation of unrealistic solutions that would minimize the physics loss function but are not physically meaningful nonetheless.
  • the inventors realized that the machine learning method may be trained to recover/reconstruct the velocities i.e. the flow profile in the flow regions outside the shadowed region using an error function for these regions as well.
  • This error function is referred to as data loss function.
  • the data loss function may not necessarily be an equation modeling a physical process, but any function suitable to train the physics informed machine learning method to reconstruct the (already known) flow profile. For example, a chi-square deviation of the estimated velocity at a location at a time point and the velocity as determined from the flow data for the same location and the same time point.
  • the method may require as an input solely a location and a time point to eventually (after training) estimate the velocity for this location at that time point.
  • the trained method is then capable to predict a velocity for any time point for virtually any location in the flow regions outside the shadowed region.
  • This training can be executed on the acquired data and does not require a specific training data set.
  • the previously mentioned physics loss-function can be constrained to solutions that connect to these boundary conditions.
  • the training of the machine learning method with regard to the physics loss function can therefore also be performed by submitting locations and time points in the shadowed region and outside of it, wherein the velocities - and as a “byproduct” the pressure governed by the physics loss function, will be determined in a way that the velocities smoothly connect to the flow profile in the flow regions.
  • a physics loss function based on a Navier-Stokes equation is sufficiently detailed and requires no further boundary or initial conditions than the ones already present in the flow regions, i.e. the velocities, that are reconstructable by means of the data loss function.
  • the method is by design capable to reconstruct flow profiles, velocities and pressure for time points that precede or follow the acquisition of the data set.
  • each function may comprise the same weight, i.e. a 1 : 1 (50:50) weighting may be applied.
  • weighting may be applied.
  • Fig. 1 shows a phantom for exemplary data acquisition
  • Fig. 2 shows a representation of acquired ultrasonic signal data in the form of one frame
  • Fig. 3 shows the pulsating flow profile of the liquid in the vessel at three different time points
  • Fig. 4 shows the capability of the method to estimate the flow profile in a shadowed region
  • Fig. 5 shows the capability of the method to determine the region occupied by the vessel
  • Fig. 6 shows a different example of reconstruction of a flow profile in a vessel comprising a stenosis in the shadowed region by way of the method according to the invention.
  • Fig. 1A shows a phantom of a polylactic acid (PI_A) vessel 1 core comprising a bifurcation for demonstration purposes.
  • the vessel 1 core is then embedded in a material providing a similar ultrasonic signal as human tissue.
  • the PI_A vessel core is then dissolved by using chloroform, and a wallless carotid bifurcation flow phantom is constructed with polyvinyl alcohol (PVA) cryogel, having diameters of 6 mm, 4.2 mm and 3.5 mm for the common, internal and external carotid artery branches, all of which are within the range of normal carotid artery sizes of adults.
  • PVA polyvinyl alcohol
  • a 50% eccentric constriction (according to the North America Symptomatic Carotid Endarterectomy Trial (NASCET) criteria) is added to construct a diseased condition with vascular stenosis.
  • NASH North America Symptomatic Carotid Endarterectomy Trial
  • Fig. 1 B shows an experimental setup for executing the method according to the invention. After the removal of this core with chloroform, the wall-less flow channel is connected to tubes 3 by hose connectors for pumping liquid through with a pump device 2.
  • Ultrasound imaging is performed by repeatedly transmitting short pulses and beamforming the returned echoes for each transmission with a transducer head 4.
  • unfocused ultrasound (plane/diverging wave) beams an image can be reconstructed by using a single pulse transmission, leading to high frame rates ( ⁇ a few KHz) useful for flow imaging.
  • focused beams the region of interest needs to be scanned sequentially with a line-by-line scheme, with a lower frame rate (determined by the density of imaging lines). Note the described flow and boundary reconstruction method is not limited to the type of ultrasound transducers and ultrasound beams adopted.
  • the UARP Ila system features a quinary excitation scheme and a harmonic reduction scheme.
  • a Verasonics L11-4v linear array transducer (Verasonics, Inc., WA, USA) is connected to the UARP Ila, and excited with a 2-cycle of sinusoidal waveform, having a center frequency of 7.55 MHz.
  • Zero-degree unfocused plane wave imaging is performed to scan the bifurcation flow phantom with a pulse repetition frequency of 6 kHz (corresponding to a 6 kHz frame rate) and a mechanical index of 0.1.
  • the ultrasound Radio Frequency (RF) data are acquired with a sampling frequency of 40 MHz.
  • the acquired RF data are transferred to a local PC 5 for beamforming and 2D motion estimation (at each pixel) by finding the frame- to- frame displacement.
  • Results and control-information may be displayed on a display 6 of the system depicted in Fig. 1 B. It is noted that the system according to the invention is typically devoid of the flow pump and the tubing as well as the phantom.
  • a delay-and-sum beamformation algorithm is used to reconstruct the ultrasound image.
  • a singular value decomposition (SVD) filter is applied to the image sequence to filter out the tissue background signal.
  • 2D vector flow mapping is then performed on the beamformed and SVD filtered RF frames with a correlation-based motion estimation method.
  • the resultant flow maps have a frame rate of 6 kHz.
  • Fig. 2A a single frame (#01131) of the recording of the wall-less flow phantom, i.e. the embedded vessel 1 , with the ultrasonic measurement system is depicted.
  • the liquid comprises tracers for enhancing the contrast of the liquid. It is noted that blood may be recorded without tracers while still providing sufficiently high contrast in the images to estimate a flow profile in the vessel.
  • Fig. 2B depicts a filtered image, in which a background signal from the tissue material, i.e. the embedding material of the vessel 1 , surrounding the vessel 1 has been removed by a software filter.
  • the flow profile obtained from the ultrasonic signal data for a single frame is depicted in Fig. 3 for three different time points.
  • Fig. 3A at time point 814
  • Fig. 3B at time point 1072
  • Fig. 3C at time point 1612.
  • a velocity vector is shown for a plurality of locations in the vessel at each time point.
  • the color-coding is indicative of the magnitude of the velocity.
  • Fig. 4A the data are prepared such that a shadow region is generated by deleting information comprised in a region located between two flow regions, one of the flow regions being located upstream the shadow region and a second flow region being located downstream the shadow region.
  • the method determines from the two flow regions the flow data, comprising the velocities at various time points and locations. Segmentation of the available vessel boundary outside the acoustic shadow may be determined as well. For the tissue background that is outside the acoustic shadow, a flow velocity value of zero is assigned (no flow there). Construction of the ⁇ -informed neural network for determination of flow based on the flow data, and i flow and vessel information.
  • the shadow region is artificially created by removing the data (tissue background & flow) beneath a vessel boundary.
  • the ground truth is available for assessing the performance of the method according to the invention.
  • the inventors leverage the capability of deep neural networks as universal function approximators to identify a nonlinear mapping between the spatio-temporal coordinates and flow parameters.
  • a deep neural network is defined, and it is trained to match the available velocity data over a period of time, whilst the solution is constrained to respect the prior computational fluid dynamics knowledge that explains the observed data.
  • the time-dependent characteristics of the flow pattern is captured by the trained neural network, and the trained neural network is then used to infer the spatio-temporal flow filed under the acoustic shadow.
  • This physics-informed neural network takes spatial (x, y) and temporal (f) coordinates as the inputs.
  • the physics-informed neural network has 10 hidden fully-connected layers, each composed of 120 neurons, and outputs two-dimensional velocity components (u, v) and pressure (p). That is to say, the physics-informed neural network maps the inputs (x, y, f) to the outputs (u, v, p).
  • the hyperbolic tangent function or a sinusoidal function is adopted as the nonlinear activation function for each neuron.
  • the flow data pixel location [x, y]
  • the corresponding temporal information for each training point (t) are fed into the neural network as inputs.
  • a velocity mismatch between its output velocity components [u, v] and the measured reference values is used.
  • the Navier-Stokes equations are valid within the flow region as the physics constraints.
  • the visible zero-velocity tissue background region is also involved in the calculation of the above velocity data mismatch, and the velocity predictions away from zero are penalized.
  • the trained neural network can be used for predictions, by feeding arbitrary spatial coordinates including those residing in the tissue background under the acoustic shadow (with predicted velocity values approaching zero). In this implementation, data of 200 ms covering the peak velocity period are used.
  • the parameters of the neural network are optimised with the Adam optimization algorithm for 20000 iterations. Starting from 1e-3, the learning rate decreased by 25% after every 1000 iterations.
  • the neural network is trained, it is used to reconstruct the flow field including the dark acoustic shadow region by feeding corresponding spatial and temporal coordinates.
  • Fig. 4A shows an example of flow data 100 with acoustic shadow region 103 (note outside the dark acoustic shadow 103 and within the background region 105, the two-dimensional velocity components are set to zero).
  • Fig. 4B shows the ground truth with all ultrasound signal available.
  • Fig. 4C is the corresponding estimation from the method according to the invention.
  • Fig. 4A depicts recorded flow data 200 representing a single time point of a recorded fluid flow through a vessel.
  • the vessel bifurcates and at the bifurcation there is a shadowed region arranged.
  • the shadowed region 103 has been added after generating the ground truth (cf. Fig. 4B) by artificially removing all flow information from this shadowed region 103.
  • Fig. 4C depicts the solution as produced by the method according to the invention, employing the deep neural network and training parameters laid out in the preceding section (10 hidden fully-connected layers, each composed of 120 neurons, using data of 200 ms covering the peak velocity period; parameters of the neural network being optimised with the Adam optimization algorithm for 20000 iterations. Starting from 1e-3, the learning rate decreased by 25% after every 1000 iterations).
  • a flow profile 104 is estimated that looks very similar to the flow profile in the recorded flow data (Fig. 4B.). While Fig. 4 shows the flow profile only for one time point, it is clear that this method is applied to all frames of the recorded flow data with comparable results.
  • the estimated velocities for the shadow region 103 are very accurate.
  • the mean error of a lateral velocity component in the shadow region is 2.43 ⁇ 0.26 cm/s.
  • Mean error of the axial velocity component in the shadow region is 2.14 ⁇ 0.35 cm/s. These values correspond to ⁇ 4 % error relative to the peak value of about 55 cm/s.
  • the gray scale is indicative of the recorded or estimated velocity in the vessel 1.
  • One additional benefit of this method is the delineation of the vessel boundary under the acoustic shadow region based on the flow velocity map that is generated by the method, which would be valuable in anatomy analysis.
  • FIG. 5 an embodiment of the invention is shown that allows to determine a vessel 1 boundary from the estimated flow profile.
  • Fig. 5A an estimated vessel 1 boundary is show, wherein the boundary has been determined by an expert.
  • Fig. 5B the boundary of the same vessel is estimated by the method, by analyzing the predicted flow profile through mathematical operations, e.g. Otsu’s method. It can be seen that the results are very similar.
  • the method allows to estimate the boundary of the vessel in the shadowed region.
  • Regions in which the vessel 1 is located are black, wherein regions where the vessel 1 does not extend to are white.
  • the boundary is at the transmission from black to white.
  • FIG. 6 another example of a reconstruction of a flow profile in a shadowed region in a vessel is depicted in analogy to Fig. 4.
  • the vessel has no bifurcation but a reduced diameter in the shadowed region, e.g. due to a stenosis.
  • Training of the physics-informed machine learning method is in essence the same as described for Fig. 4 and the sections referring to the training procedure.
  • the neural network used has the same architecture, i.e. the same number of layers and the same numbers of neurons as the one used for Fig. 4.
  • Fig. 6A shows an example of recorded flow data 100 with acoustic shadow region 103 of about 10 mm in length along the net flow direction.
  • the flow data stem from a vessel having a stenosis in the region of the shadowed region.
  • This data set has been obtained similarly to the data set of Fig. 4A, by modifying the flow data acquired from the ground truth data set and artificially applying a shadowed region.
  • the flow regions 101 , 102 up- and down-stream of the shadowed region 103 are identical to the corresponding regions in the ground truth data, which corresponds to a realistic measurement scenario.
  • Fig. 6B shows the ground truth with all ultrasound signal available.
  • Fig. 6C represents the reconstructed velocity 104 in the shadowed region using the method according to the invention.
  • Fig. 6A depicts recorded flow data 200 representing a single time point of a recorded fluid flow through the vessel.
  • the vessel does not bifurcate but exhibits a stenosis at the central region.
  • the shadowed region 103 is obtained by artificially removing all flow information from the recorded data of Fig. 6B.
  • Fig. 6C depicts the solution as produced by the method according to the invention.
  • a flow profile 104 is estimated that looks very similar to the flow profile in the recorded flow data (Fig. 6B). While Fig. 6 shows the flow profile only for one time point, it is clear that this method may be applied to all frames of the recorded flow data with comparable results.
  • the experiment in Fig. 6 is performed with a walled and stenotic PVA straight vessel phantom of 6 mm internal diameter. A pulsatile flow of 60 strokes/min and 2 mL/stroke is employed. Coherent plane wave imaging is performed with 3 steering angles of -3, 0, and 3 degrees at a pulse repetition frequency of 6 kHz, resulting in an effective frame rate of 2 kHz. All other experimental setup, beamforming, and signal conditioning steps are identical to those utilized for the bifurcation flow experiment described for Fig. 4. Motion estimation is carried out with corresponding low- resolution beamformed image pairs prior to compounding.
  • Flow data of 200 ms are processed with a fully-connected deep neural network that has an identical architecture and training configurations used for the bifurcation flow data in Fig. 4, and those have been elaborated in the preceding paragraphs.
  • the accuracy of reconstruction of the flow profile in the shadowed region in this experiment is as follows: lateral mean velocity error (across time): 1.71 ⁇ 0.16 cm/s, and axial mean velocity error (across time): 0.54 ⁇ 0.14 cm/s.

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Abstract

The invention relates to a method, particularly a computer-implemented method for estimating a flow profile of a fluid flow in a vessel comprising the steps of: - Acquiring a series of ultrasonic signal data comprising information on a fluid flow in the vessel, - Identifying a shadowed region in the ultrasonic signal data, that is devoid of information on the fluid flow in the vessel. - Determining a flow profile of the fluid from the ultrasonic signal data for flow regions of the vessel outside the shadowed region, - Estimating the flow profile in the shadowed region with a physics-informed machine learning method that is trained with flow data comprising information on the flow profile determined for the flow regions outside the shadowed region and location data comprising a plurality of locations in and/or outside the shadowed region and associated time points.

Description

Method for estimating a flow profile in a shadowed region from ultrasonic signal data
Specification
The invention relates to a method, particularly a computer-implemented method for estimating a flow profile of a fluid flow in a vessel.
Ultrasonic flow imaging has been established as a front-line tool in diagnosing a range of diseases such as vessel stenosis and congenital heart diseases, among others. Its high frame rates (> tens of Hz) provide the unique advantage of accurately capturing complex flow patterns which could be clinically informative but cannot be readily achieved by other imaging modalities such as Magnetic Resonance Imaging (MRI) and Computed Tomography (CT).
On the other hand, ultrasonic flow imaging has its specific difficulties in challenging situations, such as acoustic shadowing that obscures the visualization of the flow field and the related vessel morphology.
Stroke is one of the leading causes of mortality and disability globally together with cancer and ischaemic heart disease. Atherosclerosis, with a plaque as the main manifestation, is responsible for many stroke cases. In response to microscopic injury at the arterial wall, a biochemical signal triggered by the endothelium induces the repair with inflammatory cells and gathering of cholesterol, fats and other substances at the damaged site, with the formation of a fatty plaque protruding into the vessel lumen. The vessel luminal diameter is subsequently reduced and eventually a stenosis occurs.
Ultrasound imaging has been the mainstay for diagnosing vessel stenosis noninvasively, and it has the advantages of being non-radioactive, bed-side accessible and affordable, compared to other imaging modalities. The analysis of morphology (ratio-percent method) and Peak-Systolic-Velocity (PSV) values are two common approaches adopted to quantify the degree of stenosis by using ultrasound. And corresponding thresholds are used to evaluate the severity and inform the consequent diagnostic and therapeutic approach. However, calcifications are present in many plaques, and the calcified plaques result in an acoustic shadow that obscures the vessel lumen, inhibiting the sonographer’s ability to obtain ultrasound measurements. For example, in a study involving 400 consecutive carotid artery ultrasound scans, acoustic shadowing was found in 14.7% of the participants. Acoustic shadowing occurs when strongly-reflecting plaques on the vessel wall are insonified by ultrasound beams. This artefact is projected on regions beneath with low intensity echoes returned, causing significant problems in morphological and haemodynamic evaluation of the stenosis.
There is a great need to deal with the acoustic shadow problem as a flow profile in the shadowed region cannot be determined.
The invention is disclosed by claim 1. Advantageous embodiments are disclosed in the dependent claims.
According to a first aspect of the invention, a method, particularly a computer- implemented method for estimating a flow profile of a fluid flow in a vessel comprises the steps of:
- Acquiring a series of ultrasonic signal data comprising information on a fluid flow in the vessel, Identifying a shadowed region in the ultrasonic signal data, that is devoid of information on the fluid flow in the vessel, Determining a flow profile of the fluid from the ultrasonic signal data for flow regions of the vessel outside the shadowed region, Estimating the flow profile in the shadowed region with a physics-informed machine learning method that is trained with flow data comprising information on the flow profile determined for the flow regions outside the shadowed region and location data comprising a plurality of locations in and/or outside the shadowed region and associated time points.
The use of a physics-informed machine learning method allows an improved prediction of the flow in the shadowed region of the vessel, based on physical laws and properties that are taken into account by the physics-informed machine learning method.
Results of the method according to the first aspect reflect a more likely fluid flow in the vessel than other machine learning methods trained to predict the fluid flow in the shadowed region, as these methods may not rely on physical law and properties of the fluid and the geometry of the vessel. Particularly it is possible to also predict the temporal evolution of the fluid flow.
According to another embodiment of the invention, the flow profile is determined for at least two flow regions, wherein a first of the at least two flow regions is located upstream the shadowed region and a second of the at least two flow regions is located downstream of the shadowed region, particularly wherein the fluid flow flows from an upstream region to a downstream region of the shadowed region.
The up- and downstream regions are therefore defined by means of a net flow direction. The up and/or the downstream region may comprise one or more vessels.
According to another embodiment of the invention, the flow regions are located adjacent to the shadowed region.
According to another embodiment of the invention, the shadowed region is detected automatically by the method.
According to another embodiment of the invention, the shadowed region extends along a net flow direction up to 5 mm, 10 mm, 20 mm, 50 mm, 100 mm, 150 mm, or even up to 300 mm. Particularly the shadowed region or a vessel diameter in the shadowed region may be in the range of 0.5 mm to 50 mm, particularly in the range of 1 mm to 30 mm.
According to another embodiment of the invention, a fully-connected deep neural network is utilized to implement the physics-informed machine learning method. The fully-connected deep neural network may take spatial and time coordinates, i.e. locations and time points, as inputs and predicts multidimensional velocity components and pressure. For a two-dimensional flow profile, the inputs are spatial coordinates x, y and the corresponding time coordinate t, and the outputs are velocity components u, v, and pressure p. For a three-dimensional flow profile, there is one additional input z , the third spatial coordinate, and one additional output w , the third velocity component. The fully-connected deep neural network may comprise multiple hidden layers, with multiple neurons in each hidden layer. A sinusoidal activation function may be used for each neuron. Note that other activation functions can be used, particularly as long as they have meaningful second-order derivatives for the determination of the physics loss, as described below.
According to another embodiment of the invention, the flow data are used to train the physics-informed machine learning method to reconstruct velocities at a plurality of locations and time points to match the velocities comprised in the flow profile based on a flow data loss function, particularly wherein the flow data loss function is indicative of deviations of the reconstructed velocities and the velocities comprised in the flow profile.
According to another embodiment of the invention, the location data may be sampled in and outside the shadowed region, wherein the location data are used to train the physics-informed machine learning method to reconstruct velocities and pressure for a plurality of locations and time points in the shadowed region applying a physics loss function. By integrating physical laws, in the form of a loss function, into the learning process, the machine learning method is constrained to produce solutions that respect the underlying physics of the flow, even in the shadowed region.
According to another embodiment of the invention, training of the physics-informed machine learning method is facilitated by minimizing a composite error function comprising the flow data loss function and the physics loss function. The flow data sampled outside the shadowed region are leveraged as boundary or initial conditions to guide the solution. Traditional machine learning methods might produce nonphysical flow profiles in the shadowed region due to the lack of training data. By enforcing physical laws as a form of regularization through a loss function, solutions deviating from these laws are penalized. This physics-informed machine learning method may be trained to satisfy the governing equations universally across the domain. Thus, it can generalize to shadowed regions with no direct measurements, providing a structured way to reconstruct a coherent and physically plausible dense flow profile across the entire domain of interest, including the shadowed region.
According to another embodiment of the invention, the composite loss function and its corresponding composite loss value is minimized by a gradient descent method.
According to another embodiment of the invention, flow data are generated, particularly sampled from the flow profile determined for the flow regions, wherein the flow data comprise information on a velocity of the fluid for a plurality of locations at a plurality of associated time points.
The flow data therefore represent a spatially resolved fluid dynamics or flow profile in the vessel portions that are not shadowed, e.g. in the flow regions.
Generation of flow data from ultra-sonic signal data is known to the skilled person.
Various methods are available to determine a flow from the signal data. For example, it is possible to determine the flow profile by speckle tracking and vector Doppler methods.
In particular, the flow data are restricted to locations comprised by the flow regions. The flow data may comprise information on a velocity for one or more locations in the flow regions at one or more time points.
According to another embodiment of the invention, training of the physics-informed machine learning method comprises providing information on locations and time points from the flow data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity, wherein a flow data loss function determines a flow data loss value indicative of a deviation of the velocities comprised in the flow data from velocities estimated by the physics-informed machine learning method.
It is noted that in the context of the current specification, the term “flow data loss” and the term “data loss” may be used synonymously. The flow data comprising information on the flow profile is particularly well-suited for a physics-informed machine learning method, as fluid flow may be physically modelled by means of a dynamic information on the flow profile of the fluid.
This embodiment allows for example to train the physics-informed machine learning method to reconstruct velocities in regions outside the shadowed regions. This allows for matching the velocities and in turn contributes for example as constraints for a physics loss function that may be based on a Navier-Stokes equation.
According to another embodiment of the invention, training of the physics-informed machine-learning method comprises providing at least some locations and associated time points from the location data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity and an estimated pressure of the fluid at the location and the time point, wherein the estimated velocities and the estimated pressures are provided to a physics loss function comprising the Navier-Stokes equations, wherein the physics loss function determines a physics loss value indicative of a deviation of the velocities and the pressures estimated by the physics-informed machine-learning method from velocities and pressures that would solve the Navier-Stokes equations. Assigning the Navier-Stokes Equation to a physics loss function provides a suitable physics-informed and physics enforced loss function, such that estimations of the machine learning method are bound to physically meaningful flow profiles in the shadowed region.
According to another embodiment of the invention, training comprises a first stage, a second stage, and a third stage, wherein in the first stage, the data loss value is determined, in the second stage the physics loss value is determined, and in the third stage a composite loss value is determined from a composite loss function comprising the data loss function and the physics loss function, particularly wherein the composite loss function comprises a sum, particularly a weighted sum, of the data loss function and the physics loss function, wherein the composite loss function is minimized during training episodes.
This three-stage process allows to model a robust training process, wherein the flow data loss function and the physics loss function are separable and their contribution to the output estimate of the flow profile may be adjusted by the composite loss function.
According to another embodiment of the invention, the Navier-Stokes equations correspond to Navier-Stokes equations adjusted for incompressible fluids, conservation of momentum, and conservation of mass.
According to another embodiment of the invention, the flow profile is determined either three-dimensionally or two-dimensionally.
This allows to apply the method to various ultrasonic recording systems, which may be designed to record two-dimensional or three-dimensional flow data.
According to another embodiment of the invention, in case the flow profile is determined three-dimensionally, the physics loss function comprises physics residual functions, , fa, fa, fa, based on the terms of Navier-Stokes equations:
Figure imgf000008_0001
Figure imgf000009_0001
wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, w are velocity components of the velocity estimated by the physics-informed machine learning method along three dimensions x, y, z, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, Lphysics, and the physics loss function is determined according to:
Figure imgf000009_0002
wherein Nphysics denotes the number of locations and associated time points for which the physics loss function has been evaluated.
According to another embodiment of the invention, wherein in case the flow profile is determined two-dimensionally, the physics loss function comprises physics residual functions, , fa, fa, based on the terms of Navier-Stokes equations:
Figure imgf000009_0003
wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, are velocity components estimated by the physics-informed machine learning method along two dimensions x, y, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, LphySics, ar|d the physics loss function is determined according to:
Figure imgf000010_0001
wherein 7Vphysics denotes the number of locations and particularly associated time points for which the physics loss function has been evaluated.
According to another embodiment of the invention, in case the flow profile is determined three-dimensionally, the data loss function and the data loss value, Ldata, is determined according to:
Figure imgf000010_0002
wherein u v, w are velocity components for location Xj, yj, Zj at time point ty of the velocity estimated by the physics-informed machine learning method and wherein “reference , “reference . Reference are the corresponding velocity components of the velocity of the provided flow data, wherein Wdata, denotes the number of locations and particularly associated time points for which the data loss function has been evaluated.
According to another embodiment of the invention, in case the flow profile is determined two-dimensionally, the data loss function and the data loss value, Ldata, is determined according to:
Figure imgf000010_0003
wherein u, v are velocity components for location x,-,yy at time point ty of the velocity estimated by the physics-informed machine learning method and wherein ureference, “reference are the corresponding velocity components of the velocity of the provided flow data, wherein Ndata , denotes the number of locations and particularly associated time points for which the data loss function has been evaluated.
According to another embodiment of the invention, the composite loss value ^composite is determined according to Lcomposite = “ l^data l + ^physics | . wherein a and b are positive numbers for weighting the contributions of the physics loss value and the data loss value.
According to another embodiment of the invention, the vessel is a blood vessel in a living body.
The shadowed region may be caused by a calcified vessel wall section.
According to another embodiment of the invention, the method is executed during an ultrasonic imaging examination of a patient, wherein the training of the physics- informed machine learning method is executed for each patient at least once.
This allows to predict patient-specific flow profiles in shadowed regions.
It is noted that sufficient training data may be acquired rapidly as the regions adjacent to the shadowed region provide a temporal flow profile that may repeat itself after each heartbeat, such that acquiring the training data is facilitated with one or a few heartbeats.
The duration of acquisition may be in the range of 0.5s to 10s.
According to another embodiment of the invention, a vessel boundary is determined at least for the flow regions, wherein a velocity of the determined flow profile is set to zero at the boundary and outside the flow regions and outside the shadowed region.
This allows a robust estimation of the flow profile in the shadowed region.
According to another embodiment of the invention, a flow profile in the shadowed region is determined from the estimated velocities using the trained physics-informed machine learning method by submitting corresponding spatial and time coordinates, e.g. location data and associated time points.
According to another embodiment of the invention, a time-resolved velocity and/or a pressure distribution for at least a portion of the shadowed region, particularly for the complete shadowed region is estimated by the trained physics-informed machine learning method.
This allows for important parameters characterizing a state of vessel flow to be evaluated and determined and without the need of more complex or invasive measurement systems. According to another embodiment of the invention, the velocity and/or the pressure estimated by the trained physics-informed machine learning method is displayed, particularly wherein the estimated velocity and/or the estimated pressure is displayed in combination with the flow region and the shadowed region.
According to another embodiment of the invention, a sequence of velocities and/or pressures as estimated by the trained physics-informed machine learning method is generated, particularly in form of a movie, such that a temporal evolution of the velocity and/or the pressure, particularly a temporal evolution of the velocity distribution and/or the estimated pressure distribution is generated, wherein said sequence(s) is/are displayed together with the flow profile, particularly with the velocities in the flow regions, such that the flow regions and the shadowed region show a continuous flow profile evolving over time, particularly in form of a movie.
This allows a person to assess the dynamics in the vessel even in the shadowed region.
According to another embodiment of the invention, the flow profile in the flow regions is determined by means of tracking tracers comprised in the fluid.
Such tracers may comprise micro-bubbles.
According to another embodiment of the invention, the method comprises the step of segmenting the vessel, such that at least a boundary of the vessel, and more particularly that a lumen of the vessel is determined, wherein said step is executed together, particularly simultaneously with the step of estimating the flow profile.
According to a second aspect of the invention, a computer program comprises computer program code, that when executed on a computer causes the computer to execute the method according to one of the preceding claims.
The computer program may be stored on a non-transitory storage medium.
According to a third aspect of the invention, an ultrasonic measurement system comprises at least the following components:
An ultrasonic transducer head system configured to generate and record the ultrasonic signal data; A computer connected to the ultrasonic transducer head system and configured to receive the ultrasonic signal data, characterized in that the computer is configured to execute the method according to the first aspect of the invention or the computer program according to the second aspect of the invention.
By using flow data available in flow regions upstream and downstream an acoustic shadow region, the method aims to reconstruct the missing flow profile and vessel boundary information under the acoustic shadow, with immediate applications to the analysis of vessel stenosis.
General working principle/inventive idea
The inventors have found a surprising way to estimate a flow profile in a shadowed region, in which no flow data may be acquired from.
At the core of the invention is the physics-informed machine learning method, that by design generates solutions that obey or at least almost obey a physical principle.
With regard to the present invention, the underlying physical principle comprises the principles or laws of fluid dynamics. By penalizing the machine learning method during training according to a deviation of the underlying physical principle or law, the machine learning method is biased towards estimations that are physical meaningful, i.e. the estimates conform to a high degree to the physical principle underlying the observed natural phenomenon, here for example a fluid flow through a vessel.
This inventive concept in particular extends to the use of one or more equations that are written such that they may serve as an error function for penalizing deviations from said equation, i.e. the physically expected solution.
In this instance, the fluid flow in a region is to be determined. For this reason, the physics-informed machine-learning method comprises an error function that at least in part comprises the laws of fluid dynamics. This part is referred to as physics loss function in the context of the current specification.
However, beyond this insight the inventors were facing the problem of how to prevent the machine learning method from generating solution obeying the physics lossfunction in the shadowed region, but with non-realistic outputs, e.g. velocities and pressure.
The inventors realized that in the flow regions outside, and particularly adjacent to the shadowed region, information is present that can be used for preventing generation of unrealistic solutions that would minimize the physics loss function but are not physically meaningful nonetheless.
To bias the physics informed machine learning method to the physical meaningful solutions, the inventors realized that the machine learning method may be trained to recover/reconstruct the velocities i.e. the flow profile in the flow regions outside the shadowed region using an error function for these regions as well. This error function is referred to as data loss function. The data loss function may not necessarily be an equation modeling a physical process, but any function suitable to train the physics informed machine learning method to reconstruct the (already known) flow profile. For example, a chi-square deviation of the estimated velocity at a location at a time point and the velocity as determined from the flow data for the same location and the same time point.
For this, the method may require as an input solely a location and a time point to eventually (after training) estimate the velocity for this location at that time point. Clearly, the trained method is then capable to predict a velocity for any time point for virtually any location in the flow regions outside the shadowed region.
This training can be executed on the acquired data and does not require a specific training data set.
Now, having a created possibility to in essence generate boundary conditions for the velocity at the interface of the shadowed region and the flow regions, the previously mentioned physics loss-function can be constrained to solutions that connect to these boundary conditions. The training of the machine learning method with regard to the physics loss function can therefore also be performed by submitting locations and time points in the shadowed region and outside of it, wherein the velocities - and as a “byproduct” the pressure governed by the physics loss function, will be determined in a way that the velocities smoothly connect to the flow profile in the flow regions.
For example, a physics loss function based on a Navier-Stokes equation is sufficiently detailed and requires no further boundary or initial conditions than the ones already present in the flow regions, i.e. the velocities, that are reconstructable by means of the data loss function.
In the following figure description, it is shown, that indeed the machine learning method trained with these two error functions, allows for strikingly accurate estimations of a flow profile, velocities and or the pressure in the shadowed region of a vessel.
Furthermore, the method is by design capable to reconstruct flow profiles, velocities and pressure for time points that precede or follow the acquisition of the data set. In order to balance the influence of the data loss function versus the physics loss function, it may be advantageous to merge these two functions in a composite function where these two functions may be weighted.
For example, each function may comprise the same weight, i.e. a 1 : 1 (50:50) weighting may be applied. Alternatively, it might be advantageous to use differing weights. The skilled person may select the weights by trying out different weights or according to a performance parameter indicative of the best-suited weights.
Figure imgf000015_0001
Particularly, exemplary embodiments are described below in conjunction with the Figures. The Figures are appended to the claims and are accompanied by text explaining individual features of the shown embodiments and aspects of the present invention. Each individual feature shown in the Figures and/or mentioned in said text of the Figures may be incorporated (also in an isolated fashion) into a claim relating to the method or device according to the present invention.
Fig. 1 shows a phantom for exemplary data acquisition;
Fig. 2 shows a representation of acquired ultrasonic signal data in the form of one frame;
Fig. 3 shows the pulsating flow profile of the liquid in the vessel at three different time points;
Fig. 4 shows the capability of the method to estimate the flow profile in a shadowed region;
Fig. 5 shows the capability of the method to determine the region occupied by the vessel; and
Fig. 6 shows a different example of reconstruction of a flow profile in a vessel comprising a stenosis in the shadowed region by way of the method according to the invention.
Fig. 1A shows a phantom of a polylactic acid (PI_A) vessel 1 core comprising a bifurcation for demonstration purposes. The vessel 1 core is then embedded in a material providing a similar ultrasonic signal as human tissue. In the implementation described here, the PI_A vessel core is then dissolved by using chloroform, and a wallless carotid bifurcation flow phantom is constructed with polyvinyl alcohol (PVA) cryogel, having diameters of 6 mm, 4.2 mm and 3.5 mm for the common, internal and external carotid artery branches, all of which are within the range of normal carotid artery sizes of adults. At the entrance to the internal carotid artery branch, a 50% eccentric constriction (according to the North America Symptomatic Carotid Endarterectomy Trial (NASCET) criteria) is added to construct a diseased condition with vascular stenosis.
Fig. 1 B shows an experimental setup for executing the method according to the invention. After the removal of this core with chloroform, the wall-less flow channel is connected to tubes 3 by hose connectors for pumping liquid through with a pump device 2.
Ultrasound imaging is performed by repeatedly transmitting short pulses and beamforming the returned echoes for each transmission with a transducer head 4. With unfocused ultrasound (plane/diverging wave) beams, an image can be reconstructed by using a single pulse transmission, leading to high frame rates (~a few KHz) useful for flow imaging. With focused beams, the region of interest needs to be scanned sequentially with a line-by-line scheme, with a lower frame rate (determined by the density of imaging lines). Note the described flow and boundary reconstruction method is not limited to the type of ultrasound transducers and ultrasound beams adopted.
Experiments are performed with the manufactured phantom and a pulsatile flow of 60 stokes/min, and 3 mL/stroke generated by a pulsatile flow pump 2 (Model 1405, Harvard Apparatus, Massachusetts, United States). The flow rate is within the range of human carotid arterial flows. SonoVue microbubbles with a dilution of 1 :2000 are used as tracers for flow estimation (the proposed method is not limited to the use of microbubble contrast agents). An Ultrasound Array Research Platform Ila (UARP Ila) developed at the Ultrasonics and Embedded System Group at the University of Leeds is used for ultrasound excitation generation and data acquisition 5. The UARP Ila system features a quinary excitation scheme and a harmonic reduction scheme. A Verasonics L11-4v linear array transducer (Verasonics, Inc., WA, USA) is connected to the UARP Ila, and excited with a 2-cycle of sinusoidal waveform, having a center frequency of 7.55 MHz. Zero-degree unfocused plane wave imaging is performed to scan the bifurcation flow phantom with a pulse repetition frequency of 6 kHz (corresponding to a 6 kHz frame rate) and a mechanical index of 0.1. The ultrasound Radio Frequency (RF) data are acquired with a sampling frequency of 40 MHz. The acquired RF data are transferred to a local PC 5 for beamforming and 2D motion estimation (at each pixel) by finding the frame- to- frame displacement. Results and control-information may be displayed on a display 6 of the system depicted in Fig. 1 B. It is noted that the system according to the invention is typically devoid of the flow pump and the tubing as well as the phantom.
A delay-and-sum beamformation algorithm is used to reconstruct the ultrasound image. A singular value decomposition (SVD) filter is applied to the image sequence to filter out the tissue background signal. 2D vector flow mapping is then performed on the beamformed and SVD filtered RF frames with a correlation-based motion estimation method. The resultant flow maps have a frame rate of 6 kHz.
In Fig. 2A a single frame (#01131) of the recording of the wall-less flow phantom, i.e. the embedded vessel 1 , with the ultrasonic measurement system is depicted. In this example the liquid comprises tracers for enhancing the contrast of the liquid. It is noted that blood may be recorded without tracers while still providing sufficiently high contrast in the images to estimate a flow profile in the vessel.
Fig. 2B depicts a filtered image, in which a background signal from the tissue material, i.e. the embedding material of the vessel 1 , surrounding the vessel 1 has been removed by a software filter.
This in turn allows for determining the vessel boundary in the flow regions.
The flow profile obtained from the ultrasonic signal data for a single frame is depicted in Fig. 3 for three different time points. In Fig. 3A at time point 814, in Fig. 3B at time point 1072 and in Fig. 3C at time point 1612. Particularly, a velocity vector is shown for a plurality of locations in the vessel at each time point. The color-coding is indicative of the magnitude of the velocity.
In Fig. 4A the data are prepared such that a shadow region is generated by deleting information comprised in a region located between two flow regions, one of the flow regions being located upstream the shadow region and a second flow region being located downstream the shadow region.
The method determines from the two flow regions the flow data, comprising the velocities at various time points and locations. Segmentation of the available vessel boundary outside the acoustic shadow may be determined as well. For the tissue background that is outside the acoustic shadow, a flow velocity value of zero is assigned (no flow there). Construction of the
Figure imgf000018_0001
^-informed neural network for determination of flow based on the flow data, and i
Figure imgf000018_0002
flow and vessel
Figure imgf000018_0003
information.
In one implementation, the shadow region is artificially created by removing the data (tissue background & flow) beneath a vessel boundary. With this configuration, the ground truth is available for assessing the performance of the method according to the invention. To implement a physics-informed machine learning method, the inventors leverage the capability of deep neural networks as universal function approximators to identify a nonlinear mapping between the spatio-temporal coordinates and flow parameters. A deep neural network is defined, and it is trained to match the available velocity data over a period of time, whilst the solution is constrained to respect the prior computational fluid dynamics knowledge that explains the observed data. Under this setting, the time-dependent characteristics of the flow pattern is captured by the trained neural network, and the trained neural network is then used to infer the spatio-temporal flow filed under the acoustic shadow. This physics-informed neural network takes spatial (x, y) and temporal (f) coordinates as the inputs. In an exemplary embodiment, the physics-informed neural network has 10 hidden fully-connected layers, each composed of 120 neurons, and outputs two-dimensional velocity components (u, v) and pressure (p). That is to say, the physics-informed neural network maps the inputs (x, y, f) to the outputs (u, v, p). The hyperbolic tangent function or a sinusoidal function is adopted as the nonlinear activation function for each neuron. To train the neural network, the flow data (pixel location [x, y]), together with the corresponding temporal information for each training point (t), are fed into the neural network as inputs. To penalize the neural network, a velocity mismatch between its output velocity components [u, v] and the measured reference values is used. The Navier-Stokes equations are valid within the flow region as the physics constraints. In the training stage it is proposed to also enforce the physics constraints within the entire region of interest including the observable flow and tissue background areas. For the tissue background region, a velocity of zero is assigned. The visible zero-velocity tissue background region is also involved in the calculation of the above velocity data mismatch, and the velocity predictions away from zero are penalized. With this setup, the trained neural network can be used for predictions, by feeding arbitrary spatial coordinates including those residing in the tissue background under the acoustic shadow (with predicted velocity values approaching zero). In this implementation, data of 200 ms covering the peak velocity period are used. The parameters of the neural network are optimised with the Adam optimization algorithm for 20000 iterations. Starting from 1e-3, the learning rate decreased by 25% after every 1000 iterations.
Once the neural network is trained, it is used to reconstruct the flow field including the dark acoustic shadow region by feeding corresponding spatial and temporal coordinates.
Fig. 4A shows an example of flow data 100 with acoustic shadow region 103 (note outside the dark acoustic shadow 103 and within the background region 105, the two-dimensional velocity components are set to zero). Fig. 4B shows the ground truth with all ultrasound signal available. Fig. 4C is the corresponding estimation from the method according to the invention.
Fig. 4A depicts recorded flow data 200 representing a single time point of a recorded fluid flow through a vessel. The vessel bifurcates and at the bifurcation there is a shadowed region arranged.
The shadowed region 103 has been added after generating the ground truth (cf. Fig. 4B) by artificially removing all flow information from this shadowed region 103.
This way the ground truth is known and it may be used for benchmarking the method according to the invention.
In Fig. 4B, the recorded flow data 200 is shown without the shadowed region.
Fig. 4C depicts the solution as produced by the method according to the invention, employing the deep neural network and training parameters laid out in the preceding section (10 hidden fully-connected layers, each composed of 120 neurons, using data of 200 ms covering the peak velocity period; parameters of the neural network being optimised with the Adam optimization algorithm for 20000 iterations. Starting from 1e-3, the learning rate decreased by 25% after every 1000 iterations).
It can be seen that in the previously shadowed region of Fig. 4A, a flow profile 104 is estimated that looks very similar to the flow profile in the recorded flow data (Fig. 4B.). While Fig. 4 shows the flow profile only for one time point, it is clear that this method is applied to all frames of the recorded flow data with comparable results. The estimated velocities for the shadow region 103 are very accurate. The mean error of a lateral velocity component in the shadow region is 2.43 ± 0.26 cm/s. Mean error of the axial velocity component in the shadow region is 2.14 ± 0.35 cm/s. These values correspond to ~4 % error relative to the peak value of about 55 cm/s.
The gray scale is indicative of the recorded or estimated velocity in the vessel 1.
One additional benefit of this method is the delineation of the vessel boundary under the acoustic shadow region based on the flow velocity map that is generated by the method, which would be valuable in anatomy analysis.
In Fig. 5 an embodiment of the invention is shown that allows to determine a vessel 1 boundary from the estimated flow profile. In Fig. 5A an estimated vessel 1 boundary is show, wherein the boundary has been determined by an expert.
In Fig. 5B the boundary of the same vessel is estimated by the method, by analyzing the predicted flow profile through mathematical operations, e.g. Otsu’s method. It can be seen that the results are very similar.
It is noted that the method allows to estimate the boundary of the vessel in the shadowed region.
Regions in which the vessel 1 is located are black, wherein regions where the vessel 1 does not extend to are white. The boundary is at the transmission from black to white.
In Fig. 6, another example of a reconstruction of a flow profile in a shadowed region in a vessel is depicted in analogy to Fig. 4. In this case, the vessel has no bifurcation but a reduced diameter in the shadowed region, e.g. due to a stenosis. Training of the physics-informed machine learning method is in essence the same as described for Fig. 4 and the sections referring to the training procedure. The neural network used has the same architecture, i.e. the same number of layers and the same numbers of neurons as the one used for Fig. 4.
This example shows, that indeed the method is applicable without a priori knowledge about vessel geometry or other parameters to various situations. The high degree of alignment of reconstructed flow profile with the actual - and for the purpose of demonstrating the capability of the invention hidden flow profile for the shadowed region as well as the matching prediction of the vessel shape in the shadowed region is remarkable.
Fig. 6A shows an example of recorded flow data 100 with acoustic shadow region 103 of about 10 mm in length along the net flow direction. The flow data stem from a vessel having a stenosis in the region of the shadowed region. This data set has been obtained similarly to the data set of Fig. 4A, by modifying the flow data acquired from the ground truth data set and artificially applying a shadowed region. Thus, the flow regions 101 , 102 up- and down-stream of the shadowed region 103 are identical to the corresponding regions in the ground truth data, which corresponds to a realistic measurement scenario.
Fig. 6B shows the ground truth with all ultrasound signal available.
Fig. 6C represents the reconstructed velocity 104 in the shadowed region using the method according to the invention.
Fig. 6A depicts recorded flow data 200 representing a single time point of a recorded fluid flow through the vessel. In contrast to Fig. 4, the vessel does not bifurcate but exhibits a stenosis at the central region.
The shadowed region 103 is obtained by artificially removing all flow information from the recorded data of Fig. 6B.
This way the ground truth is known and it may be used for benchmarking the method according to the invention.
Fig. 6C depicts the solution as produced by the method according to the invention.
It can be seen that in the previously shadowed region of Fig. 6A, a flow profile 104 is estimated that looks very similar to the flow profile in the recorded flow data (Fig. 6B). While Fig. 6 shows the flow profile only for one time point, it is clear that this method may be applied to all frames of the recorded flow data with comparable results.
The specific experimental settings for the data shown in Fig. 6 are disclosed in the following.
The experiment in Fig. 6 is performed with a walled and stenotic PVA straight vessel phantom of 6 mm internal diameter. A pulsatile flow of 60 strokes/min and 2 mL/stroke is employed. Coherent plane wave imaging is performed with 3 steering angles of -3, 0, and 3 degrees at a pulse repetition frequency of 6 kHz, resulting in an effective frame rate of 2 kHz. All other experimental setup, beamforming, and signal conditioning steps are identical to those utilized for the bifurcation flow experiment described for Fig. 4. Motion estimation is carried out with corresponding low- resolution beamformed image pairs prior to compounding. Flow data of 200 ms are processed with a fully-connected deep neural network that has an identical architecture and training configurations used for the bifurcation flow data in Fig. 4, and those have been elaborated in the preceding paragraphs. The accuracy of reconstruction of the flow profile in the shadowed region in this experiment is as follows: lateral mean velocity error (across time): 1.71 ± 0.16 cm/s, and axial mean velocity error (across time): 0.54 ± 0.14 cm/s.
These results demonstrate the high accuracy and reliability of the method according to the invention.
*****

Claims

Claims
1. A method, particularly a computer-implemented method for estimating a flow profile (100) of a fluid flow in a vessel (1) comprising the steps of:
- Acquiring a series of ultrasonic signal data (200) comprising information on a fluid flow in the vessel (1), Identifying a shadowed region (103) in the ultrasonic signal data, that is devoid of information on the fluid flow in the vessel (1), Determining a flow profile (100) of the fluid from the ultrasonic signal data for flow regions (101 , 102) of the vessel (1) outside the shadowed region (103), Estimating the flow profile (104) in the shadowed region (103) with a physics-informed machine learning method that is trained with flow data comprising information on the flow profile (100) determined for the flow regions (101 , 102) outside the shadowed region (103) and location data comprising a plurality of locations in and/or outside the shadowed region (103) and associated time points.
2. The method according to one of the preceding claims, wherein the flow profile
(100) is determined for at least two flow regions (101 , 102), wherein a first
(101) of the at least two flow regions (101) is located upstream the shadowed region (103) and a second (102) of the at least two flow regions (101 , 102) is located downstream of the shadowed region (103), particularly wherein the fluid flow flows from an upstream region (101) to a downstream region (102) of the shadowed region (103).
3. The method according to one of the preceding claims, wherein flow data (200) is generated from the flow profile determined for the flow regions, wherein the flow data (200) comprises information on a velocity of the fluid for a plurality of locations at a plurality of associated time points.
4. The method according to one of the preceding claims, wherein training of the physics-informed machine learning method comprises providing information on locations and time points from the flow data (200) as input to the physics- informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity, wherein a flow data loss function determines a flow data loss value indicative of a deviation of the velocities comprised in the flow data (200) from velocities estimated by the physics-informed machine learning method.
5. The method according to one of the preceding claims, wherein training of the physics-informed machine-learning method comprises providing at least some locations and associated time points from the location data as input to the physics-informed machine learning method, wherein the physics-informed machine learning method determines for each location and time point an estimated velocity and an estimated pressure of the fluid at the location and the time point, wherein the estimated velocities and the estimated pressures are provided to a physics loss function comprising the Navier-Stokes equations, wherein the physics loss function determines a physics loss value indicative of a deviation of the velocities and the pressures estimated by the physics- informed machine-learning method from velocities and pressures that would solve the Navier-Stokes equations.
6. The method according 4 and 5, wherein training comprises a first stage, a second stage, and a third stage, wherein in the first stage, the data loss value is determined, in the second stage the physics loss value is determined, and in the third stage a composite loss value is determined from a composite loss function comprising the data loss function and the physics loss function, particularly wherein the composite loss function comprises a sum, particularly a weighted sum, of the data loss function and the physics loss function, wherein the composite loss function is minimized during training episodes.
7. The method according one of the preceding claims, wherein the flow profile (100) is determined either three-dimensionally or two-dimensionally.
8. The method according to claim 7, wherein in case the flow profile is determined three-dimensionally, the physics loss function comprises physics residual functions,
Figure imgf000024_0001
f2,f3, f4, based on the terms of Navier-Stokes equations: du dv dw fi := T d-x + T d-y + d“z
(du du du du\ dp d2u d2u d2u f2'- = \dt + dx + dy + dz) d -x dx 2z + T d-y 2z + T dz" 2z )
Figure imgf000025_0001
wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, w are velocity components of the velocity estimated by the physics-informed machine learning method along three dimensions x, y, z, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, Lphysics, and the physics loss function is determined according to:
Figure imgf000025_0002
wherein Nphysics denotes the number of locations or the number of locations and associated time points for which the physics loss function has been evaluated.
9. The method according to claim 7, wherein in case the flow profile (100) is determined two-dimensionally, the physics loss function comprises physics residual functions, f , f2,f3, based on the terms of Navier-Stokes equations:
Figure imgf000025_0003
wherein denotes a partial differential operator, t denotes time, p denotes a particularly constant fluid density, p denotes a viscosity of the fluid, u, v, are velocity components of the velocity estimated by the physics-informed machine learning method along two dimensions x, y, wherein p is the pressure as estimated by the physics-informed machine learning method, wherein the physics loss value, Lphysics, and the physics loss function is determined according to:
Figure imgf000025_0004
wherein /Vphysics denotes the number of locations or the number of locations and associated time points for which the physics loss function has been evaluated.
10. The method according to claim 7 or 8, wherein in case the flow profile (100) is determined three-dimensionally, the data loss function and the data loss value, Ldata, is determined according to:
Figure imgf000026_0001
wherein u v, w are velocity components for location Xj, yj, Zj at time point t7 of the velocity estimated by the physics-informed machine learning method and wherein ureference . ^reference . wreference are the corresponding velocity components of the velocity of the provided flow data, wherein /Vciata , denotes the number of locations or the number of locations and associated time points for which the data loss function has been evaluated.
11. The method according to claim 7 or 9, wherein in case the flow profile (100) is determined two-dimensionally, the data loss function and the data loss value, Ldata, is determined according to:
Figure imgf000026_0002
wherein u, v are velocity components for location Xj, yj at time point tj of the velocity estimated by the physics-informed machine learning method and wherein ureference, vreference are the corresponding velocity components of the velocity of the provided flow data, wherein Ndata , denotes the number of locations or the number of locations and associated time points for which the data loss function has been evaluated.
12. The method according to one of the preceding claims, wherein the method is executed during an ultrasonic imaging examination of a patient, wherein the training of the physics- informed machine learning method is executed for each patient at least once. 13. The method according to one of the preceding claims, wherein a time-resolved velocity and/or a pressure distribution for at least a portion of the shadowed region (103), particularly for the complete shadowed region is estimated by the trained physics-informed machine learning method.
14. A computer program comprising computer program code, that when executed on a computer causes the computer to execute the method according to one of the preceding claims.
15. An ultrasonic measurement system comprising at least the following components:
- An ultrasonic transducer head system configured to generate and record the ultrasonic signal data;
- A computer connected to the ultrasonic transducer head system and configured to receive the ultrasonic signal data, characterized in that the computer is configured to execute the method according to one of the claims 1 to 13 or the computer program according to claim 14.
*****
PCT/EP2024/070471 2023-07-18 2024-07-18 Method for estimating a flow profile in a shadowed region from ultrasonic signal data Pending WO2025017158A1 (en)

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